I believe the post http://mathoverflow.net/questions/198428/can-you-write-r2-as-a-disjoint-union-of-two-totally-disconnected-sets has been closed (or put "on hold") erroneously, as a duplicate of http://mathoverflow.net/questions/156/can-you-explicitly-write-r2-as-a-disjoint-union-of-two-totally-path-disconnecte
Certainly the question is different; note that the consensus to the first question is that the answer is 'no', and the answer to the second is 'yes'. In some sense the closed question was answered by Gerald Edgar at the other thread, but I think that answer needs to be revisited as being not quite a complete answer, since Włodzimierz Holsztyński has given what seems to be a valid objection in a comment below that answer, and no response to that objection was given.
Rather than have Włodzimierz respond at the path-disconnected thread (giving yet another answer to what is after all a different question), it seems to me proper to reopen the closed thread and have him and/or others answer. The question seems to me to be legitimately of MO level, even though it might be "trivial" for an expert like Włodzimierz.